RULE. Subtract the common difference from twice the first term; divide the remainder by twice the common difference; to the square of this quotient add the quotient of twice the sum of all the terms divided by the common difference; extract the square root of the sum; then divide twice the first term, diminished by the common difference, by twice the common difference, and subtract this quotient from the root just found. EXAMPLES. 1. The first term of an arithmetical progression is 7, the common difference is 1, and the sum of all the terms is 142. What is the number of terms? In this example, the common difference, subtracted from twice the first term, gives 133, which, divided by twice the common difference, gives 27, which, squared, becomes 7561. Twice the sum of all the terms, divided by the common difference, gives 1136, which, added to 7561, gives 18921, the square root of which is 431; from this, subtracting 27, we get 16 for the number of terms. 2. The first term of an arithmetical progression is 2, the common difference is 3, and the sum of all the terms is 442. What is the number of terms? Ans. 17. 3. The first term of an arithmetical progression is, the common difference is 1, and the sum of all the terms is 601. What is the number of terms? Ans. 26. Case XVIII. Given the common difference, the last term, and the sum of all the terms, to find the number of terms. RULE. To twice the last term add the common difference; divide the sum by twice the common difference; square the quotient, and from this square subtract the quotient of twice the sum of the terms divided by the common difference; extract the square root of the remainder ; then subtract this root from the quotient of the sum of twice the last term and common difference, divided by twice the common difference. EXAMPLES. 1. The common difference of the terms of an arithmetical progression is, the last term is 351, and the sum of all the terms is 1900. What is the number of terms? In this example, twice the last term, increased by the common difference, is 71, which, divided by twice the common difference, gives 107; this, squared, becomes 11449. Again, twice the sum of all the terms, divided by the common difference, gives 11400; this, subtracted from 11449, gives 49, whose square root is 7. Subtracting this root from 107, we get 100 for the number of terms. 2. The common difference of the terms of an arithmetical progression is, the last term is 37, and the sum of all the terms is 601. What is the number of terms? Ans. 26. 3. The common difference of the terms of an arithmetical progression is 1, the last term is 14, and the sum of all the terms is 105. What is the number of terms? Ans. 14. Case XIX. Given the first term, the common difference, and the sum of all the terms, to find the last term. RULE. From the first term subtract half the common difference, and to the square of the remainder add twice the product of the common difference into the sum of all the terms; then extract the square root; which, diminished by half the common difference, will give the last term. EXAMPLES. 1. The first term of an arithmetical progression is 4, the common difference is 7, and the sum of all the terms is 10233. What is the last term? 1 In this example, half the common difference, subtracted from the first term, gives, which, squared, is ; this, added to twice the product of the common difference into the sum of all the terms, which is 2×7×10233= 143262, gives 573049, whose square root is 151; from this, subtract half the common difference, and we find 137-7-750-375 for the last term. 2 2 2. The first term of an arithmetical progression is 3, the common difference is, and the sum of all the terms is 1180. What is the last term? Ans. 39. 3. A man has several sons, whose ages are in arithmetical progression; the age of the youngest is 5 years, the common difference of their ages is 6 years, and the sum of all their ages is 161. What is the age of the eldest? Ans. 41 years. Case XX. Given the common difference, the last term, and the sum of all the terms, to find the first term. RULE. Add half the common difference to the last term, and from the square of the sum subtract twice the product of the common difference into the sum of all the terms; then extract the square root of the remainder, and to this root add half the common difference. EXAMPLES. 1. The common difference of the terms of an arithmetical progression is 4, the last term is 1008, and the sum of all the terms is 127512. What is the first term? In this example, half the common difference, added to the last term, gives 1010, which, squared, is 1020100; twice the product of the common difference into the sum of all the terms is 1020096, which, subtracted from 1020100, leaves 4, the square root of which is 2; this, increased by half the common difference, becomes 4 for the first term. 2. The common difference of the terms of an arithmetical progression is 3, the last term is 49, and the sum of all the terms is 420. What is the first term? Ans. 7. 3. The common difference of the terms of an arithmetical progression is 10, the last term is 1003, and the sum of all the terms is 50800. What is the first term? Ans. 13. CHAPTER XI. GEOMETRICAL PROGRESSION. A SERIES of numbers which succeed each other regularly, by a constant multiplier, is called a geometrical · progression. This constant factor, by which the successive terms are multiplied, is called the ratio. When the ratio is greater than a unit, the series is called an ascending geometrical progression. When the ratio is less than a unit, the series is called a descending geometrical progression. Thus, 1, 3, 9, 27, 81, &c., is an ascending geometrical progression, whose ratio is 3. 1 And 1, 1, 1, 64, &c., is a descending geometrical progression, whose ratio is 1. 4 In geometrical progression, as in arithmetical progression, there are five things to be considered: 1. The first term. 2. The last term. 3. The common ratio. 4. The number of terms. 5. The sum of all the terms. These quantities are so related to each other, that any three being given, the remaining two can be found. |